$1.$ Solve Chap. $14($Multicriteria$)$ #$3($L$.$ Young company$).$ The answers including the graphs should be neat. Calculations should be shown and neat.

The L$.$ Young & Sons Manufacturing Company produces two products, which have the following profit and resource requirement characteristics:

Last month$$s production schedule used $350$ hours of labor in department A and $1000$ hours of labor in department B$.$

Young$$s management has been experiencing workforce morale and labor union problems during the past six months because of monthly departmental workload fluctuations. New hiring, layoffs, and interdepartmental transfers have been common because the firm has not attempted to stabilize workload requirements.

Management would like to develop a production schedule for the coming month that will achieve the following goals:

GOAL $1$: Use $350$ hours of labor in department A$.$

GOAL $2$: Use $1000$ hours of labor in department B$.$

GOAL $3$: Earn a profit of at least $$1300.$

a$.$ Formulate a goal programming model for this problem, assuming that goals $1$ and $2$ are P$1$ level goals and goal $3$ is a P$2$ level goal; assume that goals $1$ and $2$ are equally important.

b$.$ Solve the model formulated in part $($a$)$ using the graphical goal programming procedure.

c$.$ Suppose that the firm ignores the workload fluctuations and considers the $350$ hours in department A and the $1000$ hours in department B as the maximum available. Formulate and solve a linear programming problem to maximize profit subject to these constraints.

d$.$ Compare the solutions obtained in parts $($b$)$ and $($c$).$ Discuss which approach you favor, and why.

e$.$ Reconsider part $($a$)$ assuming that the priority level $1$ goal is goal $3$ and the priority level $2$ goals are goals $1$ and $2$; as before, assume that goals $1$ and $2$ are equally important. Solve this revised problem using the graphical goal programming procedure and compare your solution to the one obtained for the original problem.